IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. … · frequency reuse factor. Finally, Section VII is devoted to the nu-merical illustrations of our results. II. SYSTEM MODEL AND PREVIOUS

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 2, FEBRUARY 2010 735

Resource Allocation for Downlink Cellular OFDMASystems—Part II: Practical Algorithms and

Optimal Reuse FactorNassar Ksairi, Member, IEEE, Pascal Bianchi, Member, IEEE, Philippe Ciblat, Member, IEEE, and

Walid Hachem, Member, IEEE

Abstract—In a companion paper (see “Resource Allocation forDownlink Cellular OFDMA Systems—Part I: Optimal Alloca-tion,” IEEE Trans. Signal Process., vol. 58, no. 2, pp. 720–734, Feb.2010), we characterized the optimal resource allocation in termsof power control and subcarrier assignment, for a downlink sec-torized OFDMA system impaired by multicell interference. In ourmodel, the network is assumed to be one dimensional (linear) forthe sake of analysis. We also assume that a certain part of the avail-able bandwidth is likely to be reused by different base stations whilethat the other part of the bandwidth is shared in an orthogonalway between these base stations. The optimal resource allocationcharacterized in Part I is obtained by minimizing the total powerspent by the network under the constraint that all users’ raterequirements are satisfied. It is worth noting that when optimalresource allocation is used, any user receives data either in thereused bandwidth or in the protected bandwidth, but not in both(except for at most one pivot-user in each cell). We also proposedan algorithm that determines the optimal values of users’ resourceallocation parameters. As a matter of fact, the optimal allocationalgorithm proposed in Part I requires a large number of operations.In the present paper, we propose a distributed practical resourceallocation algorithm with low complexity. We study the asymptoticbehavior of both this simplified resource allocation algorithm andthe optimal resource allocation algorithm of Part I as the numberof users in each cell tends to infinity. Our analysis allows to provethat the proposed simplified algorithm is asymptotically optimal,i.e., it achieves the same asymptotic transmit power as the optimalalgorithm as the number of users in each cell tends to infinity. As abyproduct of our analysis, we characterize the optimal value of thefrequency reuse factor. Simulations sustain our claims and showthat substantial performance improvements are obtained whenthe optimal value of the frequency reuse factor is used.

Index Terms—Asymptotic analysis, distributed resource alloca-tion, multicell resource allocation, OFDMA.

I. INTRODUCTION

I N a companion paper [1], we introduced the problem ofjoint power control and subcarrier assignment in the down-

link of a one-dimensional sectorized two-cell OFDMA system.

Manuscript received October 31, 2008; accepted August 08, 2009. First pub-lished September 29, 2009; current version published January 13, 2010. Theassociate editor coordinating the review of this manuscript and approving it forpublication was Dr. Biao Chen.

N. Ksairi is with Supélec, 91192 Gif-sur-Yvette Cedex, France (e-mail:[email protected]).

P. Bianchi, P. Ciblat, and W. Hachem are with the CNRS/Telecom ParisTech(ENST), 75634 Paris Cedex 13, France (e-mail: [email protected];[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2009.2033307

Resource allocation parameters have been characterized in sucha way that i) the total transmit power of the network is minimumand ii) all users’ rate requirements are satisfied. Similarly to [2],we investigate the case where the channel state information atthe base station (BS) side is limited to some channel statistics.However, contrary to [2], our model assumes that the availablebandwidth is divided into two bands: The first one is reused bydifferent base stations (and is thus subject to multicell interfer-ence) while the second one is shared in an orthogonal way be-tween the adjacent base stations (and is thus protected from mul-ticell interference). The number of subcarriers in each band isdirectly related to the frequency reuse factor. We also assumethat each user is likely to modulate subcarriers in each of thesetwo bands and thus we do not assume a priori a geographicalseparation of users modulating in the two different bands. Thesolution to the above resource allocation problem is given in thefirst part of this work. This solution turns out to be “binary”: ex-cept for at most one pivot-user, users in each cell must be dividedinto two groups, the nearest users modulating subcarriers onlyin the reused band and the farthest users modulating subcarriersonly in the protected band. An algorithm that determines theoptimal values of users’ resource allocation parameters is alsoproposed in the first part.

It is worth noting that this optimal allocation algorithm isstill computationally demanding, especially when the number ofusers in each cell is large. One of the computationally costliestoperations involved in the optimal allocation is the determi-nation of the pivot-user in each cell. In the present paper, wepropose a distributed simplified resource allocation algorithmwith low computational complexity, and we discuss its perfor-mance as compared to the optimal resource allocation algorithmof Part I. This simplified algorithm assumes a pivot-distancethat is fixed in advance prior to the resource allocation process.Of course, this predefined pivot-distance should be relevantlychosen. For that sake, we show that when the fixed pivot-dis-tance of the simplified algorithm is chosen according to a certainasymptotic analysis of the optimal allocation scheme, the per-formance of the simplified algorithm is close to the optimal one,provided that the number of users in the network is large enough.Therefore, following the approach of [2], we propose to charac-terize the limit of the total transmit power which results from theoptimal resource allocation policy as the number of users in eachcell tends to infinity. Several existing works on resource alloca-tion resorted to this kind of asymptotic analysis, principally inorder to get tractable formulations of the optimization problem

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736 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 2, FEBRUARY 2010

Fig. 1. Two-cell system model.

that can be solved analytically. For example, the asymptoticanalysis was used in [3] and [4] in the context of downlink anduplink single cell OFDMA systems respectively, as well as in [5]in the context of code-division multiple-access (CDMA) sys-tems with fading channels. Another application of the asymp-totic analysis can be found in [6]. The authors of the cited workaddressed the optimization of the sum rate performance in amulticell network. In this context, the authors proposed a decen-tralized algorithm that maximizes an upper-bound on the net-work sum rate. Interestingly, this upper-bound is proved to betight in the asymptotic regime when the number of users per cellis allowed to grow to infinity. However, the proposed algorithmdoes not guaranty fairness among the different users.

In this paper, we use the asymptotic analysis in order to obtaina compact form of the (asymptotic) power transmitted by thenetwork for the optimal resource allocation algorithm, and weuse this result to propose relevant values of the fixed pivot-dis-tance associated with the simplified allocation algorithm. Weprove in particular that when this fixed pivot-distance is chosenequal to the asymptotic optimal pivot-distance, then the powertransmitted when using the proposed simplified resource allo-cation is asymptotically equivalent to the minimum power as-sociated with the optimal algorithm. This limiting expressionno longer depends on the particular network configuration, buton an asymptotic, or “average,” state of the network. More pre-cisely, the asymptotic transmit power depends on the averagerate requirement and on the density of users in each cell. Italso depends on the value of the frequency reuse factor. Asa byproduct of our asymptotic analysis, we are therefore able todetermine an optimal value of the latter reuse factor. This op-timal value is defined as the value of which minimizes theasymptotic power.

The rest of this paper is organized as follows. In Section IIwe recall the system model as well as the joint resource allo-cation problem. In Section III, we propose a novel suboptimaldistributed resource allocation algorithm. Section IV is devotedto the asymptotic analysis of the performance of this simplifiedallocation algorithm as well as the performance of the optimalresource allocation scheme of Part I when the number of users

tends to infinity. Theorem 1 characterizes the asymptotic be-havior of the optimal joint allocation scheme. The results of thistheorem are used in Section IV-D in order to determine relevantvalues of the fixed pivot-distances associated with the simpli-fied allocation algorithm. Provided that these relevant values areused, Proposition 2 states that the simplified algorithm is asymp-totically optimal. Section VI addresses the selection of the bestfrequency reuse factor. Finally, Section VII is devoted to the nu-merical illustrations of our results.

II. SYSTEM MODEL AND PREVIOUS RESULTS

A. System Model

We consider a sectorized downlink OFDMA cellular net-work. We focus on two neighboring one-dimensional (linear)cells, say Cell and Cell , as illustrated by Fig. 1. Denoteby the radius of each cell. We denote by the number ofusers of Cell and by the number of users of Cell . Thetotal number of available subcarriers in the system is denotedby . For a given user in Cell ,we denote by the distance that separates him/her from BS ,and by the set of indexes corresponding to the subcarriersmodulated by . is a subset of . The signalreceived by user at the th subcarrier and at the

th OFDM block is given by

(1)

where represents the data symbol transmitted by BS. Process is an additive noise which encompasses

the thermal noise and the possible multicell interference. Co-efficient is the frequency response of the channel atthe subcarrier and the OFDM block . Random variables

are assumed Rayleigh distributed with variance. Channel coefficients are supposed to be per-

fectly known at the receiver side, and unknown at the BS side.We assume that vanishes with the distance based on agiven path loss model. The set of available subcarriers is par-titioned into three subsets: containing the reused subcarriersshared by the two cells; and containing the protected


KSAIRI et al.: RESOURCE ALLOCATION FOR DOWNLINK CELLULAR OFDMA SYSTEMS—PART II 737

subcarriers only used by users in Cell and respectively.The reuse factor is defined as the ratio between the numberof reused subcarriers and the total number of subcarriers:

so that contains subcarriers. If user modulates a sub-carrier , the additive noise contains both thermal noiseof variance and interference. Therefore, the variance ofthis noise-plus-interference process depends on and coincideswith , where representsthe channel between BS and user of Cell at frequencyand OFDM block , and where is theaverage power transmitted by BS in the interference band-width . The remaining subcarriers are shared by thetwo cells, Cell and , in an orthogonal way. If user mod-ulates such a subcarrier , the additive noisecontains only thermal noise. In other words, subcarrier doesnot suffer from multicell interference. Then we simply write

. The resource allocation parameters foruser are: the power transmitted on each of the subcar-riers of the nonprotected band allocated to him, his shareof , the power transmitted on each of the subcarriers ofthe protected band allocated to him and his share of .In other words,

As a consequence, andfor each cell . Moreover, let (respectively, ) be thechannel gain-to-noise ratio (GNR) in band (respectively, ),namely (respectively, ). “Setting aresource allocation for Cell ” means setting a value for param-eters .

B. Joint Resource Allocation for Cells and

Assume that each user has a rate requirement ofnats/s/Hz. In the first Part of this work [1], our aim was tojointly optimize the resource allocation for the two cells whichi) allows to satisfy all target rates of all users, and ii)minimizes the power used by the two base stations in orderto achieve these rates. For each Cell , denote bythe adjacent cell ( and ). The ergodic capacityassociated with a user in Cell is given by

(2)

where is a standard exponentially distributed random vari-able, and where coefficient is given by

(3)

where represents the channel between BS and userof Cell at frequency and OFDM block . Coefficient

represents the signal to interference plus noise ratioin the interference band . We assume that users are numbered

from the nearest to the BS to the farthest. As in [1], the fol-lowing problem will be referred to as the joint resource allo-cation problem for Cells and : Minimize the total powerspent by both base stations

with respect to under

the following constraint that all users’ rate requirements aresatisfied i.e., for each user in any Cell , . The solu-tion to this problem has been determined in the first part of thiswork [1]. As a noticeable point, the results of [1] indicate the ex-istence in each cell of a pivot-user that separates two groups ofusers: The “protected” users and the “nonprotected” users. Thefollowing proposition states this binary property of the solution.

Proposition 1 [1]: Any global solution to the joint resourceallocation problem is “binary” i.e., there exists a user in eachCell such that for closest users , andfor farthest users .

In the sequel, we denote by the position of the pivot-user in Cell i.e., . A resource allocation algo-rithm is also proposed in [1]. This algorithm turns out to have ahigh computational complexity and the determination of the op-timal value of the pivot-distance turns out to be one of thecostliest operations involved in this algorithm. This is why wepropose in the following section of the present paper a subop-timal simplified allocation algorithm that assumes a predefinedpivot-distance.

III. PRACTICAL RESOURCE ALLOCATION ALGORITHM

A. Motivations and Main idea

Proposition 1 provides the general form of the optimal re-source allocation, showing in particular the existence of pivot-users , in both Cells , , separating the users whomodulate in band from the users who modulate in bandsand . As a matter of fact, the determination of pivot-users

is one of the costliest operations of this optimal allo-cation (see [1] for a detailed computational complexity anal-ysis). Thus, it would be convenient to propose an allocationprocedure for which the pivot-position would be fixed in ad-vance to a constant rather than systematically computed/opti-mized. We propose a simplified resource allocation algorithmbased on this idea. Furthermore, we prove that when the valueof the fixed pivot-distances is relevantly chosen, the proposedalgorithm is asymptotically optimal as the number of users in-creases. In other words, the total power spent by the network forlarge when using our suboptimal algorithm does not exceedthe minimum power that would have been spent by using theoptimal resource allocation. The proposed algorithm is basedon the following idea.

Recall the definition of and as the respec-tive position of the optimal pivot-users and defined byProposition 1. As the optimal pivot-positions andare difficult to compute explicitly and depend on the particularrates and users’ positions, we propose to replace and

with predefined values and fixed beforethe resource allocation process. In our suboptimal algorithm, allusers in Cell whose distance to the BS is less thanmodulate in the interference band . Users farther thanmodulate in the protected band . Of course, we still need to



determine the pivot-distances and . A procedurethat permits the relevant selection of is givenin Section IV-C.

B. Detailed Description

Assume that the values of and have been fixedbeforehand prior to the resource allocation process. For eachCell , define by the subset of corresponding tothe users whose distance to BS is less than . Define by

the set of users whose distance to BS is larger than .1) Resource Allocation for Protected Users: Focus for in-

stance on Cell . For each , we arbitrarily seti.e., user is forced to modulate in the protected band

only. For such users, the remaining resource allocation pa-rameters are obtained by solving the following clas-sical single cell problem w.r.t. :

“Minimize the transmitted power

under rate constraint for each .”

The above problem is a simple particular case of the singlecell problem addressed in [1]. Define the functions

andon . The solution is given by

where parameter is obtained by writing that constraintholds or equivalently, is the unique

solution to

We proceed similarly for Cell .2) Resource Allocation for Interfering Users: We now focus

on users for each Cell . For such users, wearbitrarily set i.e., users in are forced tomodulate in the interference band only, for each Cell . Theremaining resource allocation parametersare obtained by solving the following simplified multicellproblem.

Problem 1. [Multicell]: Minimize

w.r.t. under the following constraintsfor each Cell :

Clearly, the above Problem can be interpreted as a particularcase of the initial resource allocation (Problem 2 in [1]) ad-dressed in Section II-B of the present paper. The main differenceis that the initial multicell problem jointly involves the resourceallocation parameters in three bands , and whereas the

present problem only optimizes the resource allocation parame-ters corresponding to band , while arbitrarily setting the othersto zero. Therefore, the results of Part I [1], Theorem 2 of [1] inparticular, can directly be used to determine the global solutionto Problem 1.

Remark 1 (Feasibility): Recall that the initial joint resourceallocation Problem (Problem 2 in [1]) described in Section II-Bin the present paper was always feasible. Intuitively, this was dueto the fact that any user was likely to modulate in the protectedband if needed, so that any rate requirement was likely to besatisfied by simply increasing the power in the protected band.In the present case, the protected band is by definition forbiddento users in . Theoretically speaking, Problem 1 might not befeasible due to multicell interference. Fortunately, we will seethat this case does not happen, at least for a sufficiently largenumber of users, if the values of the pivot-distances and

are well chosen. This point will be discussed in moredetail in Section V.

Define as the average power trans-mitted by BS in the interference bandwidth . By straightfor-ward application of Theorem 2, we obtain that for each Celland for each user ,

(4)

(5)

where for each and for a fixed value of , parame-ters are the unique solution to the following system ofequations:

(6)

(7)

Note that the first equation is nothing else than the constraint. The second equation is nothing else than

the definition . We now prove that thesystem of four equations (6), (7) for admits a uniquesolution and we provide a simple algorithmallowing to determine this solution.

Focus on a given Cell and consider any fixed value .Denote by the RHS of (7) where is defined as theunique solution to (6). Since (7) should be satisfied for both

and , the following two equations hold

The couple is therefore clearly a fixed point of thevector-valued function :

(8)

As a matter of fact, it can be shown that such a fixed point of isunique. This claim can be proved using the approach previouslyproposed by [12].



Lemma 1: Function is such that the following propertieshold.

1) Positivity: .2) Monotonicity: If , , then

.3) Scalability: for all , .The proof of Lemma 1 uses arguments which are very similar

to the proof of Theorem 1 in [11]. It is thus omitted from thispaper and provided in [13]. Function is then a standard inter-ference function, using the terminology of [12]. Therefore, asstated in [12], such a function admits at most one fixed point.On the other hand, the existence of a fixed point is ensured bythe feasibility of Problem 1 and by the fact that (8) holds forany global solution. In other words, if Problem 1 is feasible,then function does admit a fixed point and this fixed point isunique. In the latter case, the results of [12] state furthermorethat a simple fixed point algorithm applied to function con-verges necessarily to its unique fixed point.

In practice, resource allocation in band can thus be achievedby the following procedure.

Ping-pong algorithm for interfering users:1) Initialization: .2) Cell A: Given the current value of the power trans-

mitted by base station B in the interference bandwidth,compute as the unique solution to (6), (7) with

.3) Cell B: Given the current value of , compute

by (6), (7).4) Go back to step 2 until convergence.5) Define resource allocation parameters by (4), (5).Comments:

1) Convergence of the ping-pong algorithm. We statedearlier that Problem 1 is either feasible or infeasible,depending on the value of . If the latterproblem is feasible, then function will have a uniquefixed point due to Lemma 1 and the ping-pong algo-rithm will converge to this fixed point. If Problem 1 isinfeasible, then function will have no fixed points andthe ping-pong algorithm will diverge. One of the mainpurposes of Section IV-C is to provide relevant values of

such that convergence of the ping-pongalgorithm holds for sufficiently large number of users.

2) Note that the only information needed by Base Stationabout Cell is the current value of the power trans-mitted by Base Station in the interference band . Thisvalue can i) either be measured by Base Station at eachiteration of the ping-pong algorithm, or ii) it can be com-municated to it by Base Station over a dedicated link. Inthe first case, no message passing is required, and in thesecond case only few information is exchanged betweenthe base stations. The ping-pong algorithm can thus be im-plemented in a distributed fashion.

C. Complexity Analysis

We showed earlier that allocation for protected userscan be reduced to the determination in each cell of thevalue of , which is the unique solution to the equation

. We argued in [1]that solving this kind of equations requires a computationalcomplexity proportional to the number of terms in the LHSof the equation, which is itself of order . Using similararguments, we can show that each iteration of the ping-pongalgorithm for nonprotected users can be performed with acomplexity of order . Let designate the number ofiterations needed till convergence. The overall computationalcomplexity of the ping-pong algorithm, and hence of thesimplified resource allocation scheme as well, is thus of theorder of . Our simulations showed that the ping-pongalgorithm converges relatively quickly in most of the cases.Indeed, no more than iterations were needed in almostall the simulations settings to reach convergence within a veryreasonable accuracy. The complexity of the simplified algo-rithm is to be compared with the computational complexity ofthe optimal algorithm which was shown in [1] to be of the orderof , where is the number of points inside acertain 2-D search grid.

IV. ASYMPTOTIC OPTIMALITY OF THE SIMPLIFIED RESOURCE

ALLOCATION SCHEME

The aim of this section is to evaluate the performance of theproposed simplified algorithm. The relevant performance metricin the context of this paper is the total power that must be trans-mitted by the base stations. Since the simplified algorithm as-sumes predefined pivot-distances fixed priorto the resource allocation process, the performance of the pro-posed algorithm depends on the choice of these fixed pivot-distances. One must therefore determine what relevant valueshould be selected for . A possible method isaddressed in this section and consists in studying the case wherethe number of users tends to infinity.

A. Main Tools: Asymptotic Analysis

We study first the performance of the optimal allocation al-gorithm proposed in Part I [1] when the number of users in eachcell tends to infinity. From the results of this asymptotic study,we conclude the asymptotic behavior of the optimal pivot-dis-tances . It turns out that when the number ofusers increases, the optimal pivot-distances as well as the totaltransmitted power no longer depend on the particular cell con-figuration, but on an asymptotic state of the network, such as theaverage rate requirement and the density of users in each cell.Thanks to this result, we can now choose the fixed pivot-dis-tances associated with the simplified algorithm to be equal tothe asymptotic pivot-distances. In this case, one can show thatthe performance gap between the simplified and the optimal al-location schemes vanishes for high numbers of users. We intro-duce now the mathematical assumptions and tools that we usefor defining the asymptotic regime.

1) Notations and Basic Assumptions: In the sequel, we de-note by the total bandwidth of the system in Hz. We con-sider the asymptotic regime where the number of users in eachcell tends to infinity. We denote by the data rate re-quirement of user in nats/s, and we recall that is the datarate requirement of user in nats/s/Hz. Notice that the total rate

which should be delivered by BS tends to infinity



as well. Thus, we need to let the bandwidth grow to infinityin order to satisfy the growing data rate requirement. Recallingthat denotes the total number of users in bothcells, the asymptotic regime will be characterized by ,

and where is a positive real number. We as-sume on the other hand that tends to somepositive constant as tends to infinity. Without restrictions,this constant is assumed in the sequel to be equal to 1/2 i.e.,the number of users becomes equivalent in each cell. In order tosimplify the proofs of our results, we assume without restrictionthat for each , the rate requirement is upper-bounded by acertain constant , , where can be chosen aslarge as needed, and that users of each cell are located in the in-terval where can be chosen as small as needed. Re-call that denotes the position of each user i.e., the distancebetween the user and the BS. The variance of the channel gainof user will be written as where models thepath loss. Typically, function has the formwhere is a certain gain and where is the path-loss coeffi-cient, . In the sequel, we denote bythe received gain to noise ratio in the protected bandwidth, fora user at position . This way, . Similarly, we de-fine for each user in cell , . Moregenerally, denotes the gain-to-interference-plus-noiseratio in the interference bandwidth at position when the inter-fering cell is transmitting with power in band . Functions

and are assumed to be continuous functions of. It is worth noting that for each , . Finally,

recall that coefficient (respectively, ) is defined as theratio between the part of the interference bandwidth (respec-tively, protected bandwidth ) and the total bandwidth. Thus,

and tend to zero as the total bandwidth tends to in-finity for each .

2) Statistical Tools and Main Ideas of the Asymptotic Study:Theorem 2 of Part I [1] reduces the determination of the wholeset of resource allocation parameters in both cells to the determi-nation of ten unknown parameters .Parameter in particular represents the power transmitted byCell in the nonprotected band . Consider now one of the twoCells , and denote by the second (adjacent) cell.In the sequel, we use the notation (respectively, )instead of (respectively, ) to designate the power trans-mitted by BS in the nonprotected band (respectively, theprotected band ) when the optimal solution characterized byProposition 1 is used:

(9)

(10)

The new notation is used to indicate the depen-dency of the results on the number of users . For the samereason, parameters will be denoted in the sequelby , , , respectively. Our goal now isto characterize the behavior of the resource allocation strategyas and, in particular, the behavior of powers ,

. By straightforward application of Theorem 2 of Part I,can be written as

(11)

where denotes the power trans-

mitted to the pivot-user in the interference band , andwhere function is defined by

(12)

for each . The first term in the rhs of (11) representsthe total power allocated to all users . It is quite intu-itive that the power allocated to one user is negligible

when compared to the power allocated to all users .Indeed, it will be shown in Appendix A that the first term of(11) is bounded as whereas tends to zero. Inthe sequel, we use notation , wherestands for any term which converges to zero as . Inorder to study the limit of this expression as tends to infinity,we introduce for each one of the two cells the following mea-sure defined on the Borel sets of as follows:

(13)

where and are any intervals of and where isthe Dirac measure at point . In order to have more in-sights on the meaning of this tool, it is useful to remark that

is equal to [see the equation at the bottom of thepage]. Thus, measure can be interpreted as the distribu-tion of the set of couples of Cell . The introduction ofthe above measure simplifies considerably the asymptotic studyof the transmit power. Indeed, replacing (in nats/s/Hz) by

in (11), we obtain

(14)



where integration is considered with respect to the set, where is the position of

pivot-user and where can be chosen, as stated earlierin this section, as small as needed. It is quite intuitive that theasymptotic power can be obtained from (14)by replacing by and thedistribution by the asymptotic distribution of couples

as tends to infinity. The existence and the defini-tion of this asymptotic distribution is provided by the followingassumption.

Assumption 1: As tends to infinity, measure con-verges weakly to a measure .

We refer to [7] for the materials on the conver-gence of measures. In order to have some insighton the behavior of (14) in the asymptotic regime,imagine for the sake of simplicity that sequences

are convergent and that they converge respectively to. This assumption is of course

arbitrary for the moment, but it allows to better understandthe main ideas of our asymptotic analysis. More rigorousconsiderations on the convergence of these sequences will bediscussed later on. Ignoring at first such technical issues, itis intuitive from (14) that converges to a constantdefined by

(15)

where . In other words, we manage toexpress the limit of the power transmitted by station inthe interference band as a function of the asymptotic cell con-figuration. In order to further simplify the above expression, it isalso realistic to assume that measure is the measure productof a limit rate distribution times a limit location distribution. As-sumption 2 below is motivated by the observation that in prac-tice, the rate requirement of a given user is usually not relatedto the position of the user in each cell.

Assumption 2: Measure is such thatwhere is the limit distribution of rates and is the

limit distribution of the users’ locations. Here denotes theproduct of measures.

Measures and respectively correspond to the distribu-tions of the rates and the positions of the users within one cell.For instance, the value represents theaverage rate requirement per channel use in Cell . We further-more assume that measures and are absolutely contin-uous with respect to the Lebesgue measure on . Using As-sumption 2, (15) becomes

(16)

Of course, a similar result can be obtained for i.e., thepower transmitted by base station in the protected band . Tothat end, we simply note that function satisfies

. Using similar tools, the expression of given by(24) converges as toward

(17)

Equations (16) and (17) respectively provide the limitsof and as a function of some parameters

and (assumed for the moment to be the limits ofand as long as such limits exist).

These unknown parameters still need to be characterized.Therefore, we must determine a system of equations which issatisfied by these parameters. This task is done by Theorem 1given below.

B. Asymptotic Performance of the Optimal Allocation

Define the following functionfor each

and let designate the function defined on as. The proof of the following

result is provided in Appendix A.Theorem 1: Assume that in such

a way that and . Assume thatthe optimal solution for the joint resource allocation problem(Problem 2 in [1]) is used for each . The total power spentby the networkconverges to a constant . The limit has the followingform:

(18)

where for each , the following system of equations invariables is satisfied:

(19)

(20)

(21)

(22)

Moreover, for each and for any arbitrary fixed value, the system of (19), (20), (21), (22) admits at most

one solution .As a consequence, when optimal multicell resource allocation

is used, the total power spent by the network converges to a con-stant which can be evaluated through the results of Theorem 1.This result allows to evaluate the asymptotic power spent by the



network as a function of the reuse factor , the average rate re-quirement and the asymptotic distribution of users in eachcell .

Now that the asymptotic performance of the optimal alloca-tion scheme has been studied, the value of the fixed pivot-dis-tances associated with the simplified allocationalgorithm can be relevantly chosen to be equal in each Cell tothe asymptotic pivot distance defined by Theorem 1.

C. Determination of the Fixed Pivot-Distancesfor the Simplified Allocation Scheme

We stated earlier in Section III that the suboptimal algo-rithm replaces the optimal value of the pivot-distancein each Cell with a fixed value . Intuitively, if

and are chosen such thatand for large , the performance of ouralgorithm shall be close to the optimal one as increases.Therefore, we must determine an asymptotically optimal pair ofpivot-distances . To that end we propose the followingprocedure.

Note first by referring to Theorem 1 that the value ofcan be easily determined once the relevant values of and

have been determined. The remaining task is thus the deter-mination of the value of . To that end, we propose toperform an exhaustive search on .

i) For each point on a certain 2-D search grid,solve the system (19), (20), (21), (22) introduced byTheorem 1 for both . Theorem 1 states that thissystem admits at most one solution for any arbitrary fixedvalue . If the investigated point ofthe grid is such that the system (19), (20), (21), (22) doesadmit a solution, we can obtain this solution denoted by

, , ,thanks to a simple procedure inspired by the single-cellprocedure proposed in Part I [1] for finite number ofusers:• Solve the system (19), (20), (21), (22), (22a) formed

by replacing the equality in (22) of system (19), (20),(21), (22) by the following inequality

(22a)

The existence and the uniqueness of the solution to thisnew system for an arbitrary can beproved by extending, to the case of infinite number ofusers, Proposition 1 which was provided in [1] for thecase of finite number of users.

• If the resulting powertransmitted in the interference band is equal to ,then the resulting value of coincides withthe unique solution to system (19), (20), (21), (22).Once again, this claim can be proved by extendingProposition 1 of [1] to the case of infinite number ofusers.

• If the power is lessthan , then is clearly not a solution

to system (19), (20), (21), (22), as equality (22) doesnot hold. In this case, it can be easily shown thatsystem (19), (20), (21), (22) has no solution. The point

is thus eliminated.ii) Compute the total power

that would be transmitted if the values of andintroduced by Theorem 1 were respectively equal toand .

iii) The final value of is given by ,, the value associated with the

argument of the minimum power transmitted by thenetwork:

iv) Finally, we choose

Note that the same procedure provides as a byproduct the limitof the total transmit power as .

Comments: It is clear from our previous discussion that theabove procedure for computing can be done in ad-vance prior to resource allocation. This is essentially due tothe fact that the asymptotically optimal pair of pivot-distances

does not depend on the particular cell configuration,but on an asymptotic or “average” state of the network. The pro-cedure can be run for instance before base stations are broughtinto operation. It can also be done once in a while as the asymp-totic distribution of the users and the average rate requirementcan be subject to changes: But these changes occur after longperiods of time. Therefore, the number of operations neededfor the computation of is not a major concern be-cause it does not affect the computational complexity of re-source allocation.

D. Asymptotic Performance of the Simplified Algorithm

Denote by the total power transmitted when our sim-

plified allocation algorithm is applied. Recall that desig-nates the total power transmitted by the network when the op-timal resource allocation associated with the joint resource al-location problem (Problem 2 of [1]) is used.

Proposition 2: The following equality holds:

Proposition 2 can be proved using the same arguments as theones used in Appendix A. The detailed proof is omitted. Theabove Proposition states that the proposed suboptimal algorithmtends to be optimal w.r.t. the joint resource allocation problem,as the number of users increases. Therefore, our algorithm is atthe same time much simpler than the initial optimal resourceallocation algorithm of [1], and has similar performance at leastfor a sufficient number of users in each cell. Section VII will



furthermore indicate that even for a moderate number of users,our suboptimal algorithm is actually nearly optimal.

V. ON THE CONVERGENCE OF THE SIMPLIFIED

ALLOCATION ALGORITHM

As stated before, the simplified algorithm performs the re-source allocation in each Cell independently for the protected

and the nonprotected users, which are separated bythe predefined pivot-distance . Resource allocation forthe nonprotected users is done by the iterative and distributedping-pong algorithm described in Section III. It was stated inSection III that the convergence of the ping-pong algorithm isensured by the feasibility of the problem of resource allocationfor the nonprotected users (Problem 1). If Problem 1is feasible, the ping-pong algorithm converges. If Problem 1 isinfeasible, the ping-pong algorithm diverges. It was also statedin Section III that Problem 1 may not be feasible if arbitraryvalues of the pivot-distances and are used.Fortunately, feasibility of the latter problem will not be an issueif the value of and are relevantly chosen asdescribed by the procedure introduced in Section IV-D. Indeed,it can be shown in this case that at least for large , the setwill contain the users who would anyway have been restrictedto the interference band if the optimal resource allocationof Part I [1] was used. More precisely, it can be shown thatthere exists a value of beyond which Problem 1 isalways feasible. The proof of this statement is provided in [13].It is based on sensitivity analysis of perturbed optimizationproblems [14]. It is worth mentioning that in our simulations,Problem 1 was feasible in almost all the settings of the system,even for a moderate number of users per cell as small as 25.

VI. SELECTION OF THE BEST REUSE FACTOR

The selection of a relevant value allowing to optimize thenetwork performance is of crucial importance as far as cellularnetwork design is concerned. The definition of an optimal reusefactor requires however some care. The first intuition wouldconsist in searching for the value of which minimizes the totalpower transmitted by the network, for a finitenumber of users . However, depends on the partic-ular target rates and the particular positions of users. In practice,the reuse factor should be fixed prior to the resource allocationprocess and its value should be independent of the particularcells configurations. A solution adopted by several works in theliterature consists in performing system level simulations andchoosing the corresponding value of that results in the bestaverage performance. In this context, we cite [8]–[10] withoutbeing exclusive. In this paper, we are interested in providing an-alytical methods that permit to choose a relevant value of thereuse factor. This is why we propose to select the value ofthe reuse factor as

Recall that the limiting power is givenby (18). In practice, we propose to compute the value of

for several values of on a grid in the interval [0,1]. Foreach value of on the grid, can be obtained using the

procedure presented in Section IV-C. Note also that complexityissues are of few importance, as the optimization is done priorto the resource allocation process. It does not affect the com-plexity of the global resource allocation procedure. We shall seein Section VII that significant gains are obtained when using theoptimized value of the reuse factor instead of an arbitrary value.

VII. SIMULATIONS

We first begin by presenting the technical parameters of thesystem model. In our simulations, we considered a free spaceloss (FSL) model characterized by a path loss exponent aswell as the so-called Okumura-Hata (O-H) model for open areas[15] with a path loss exponent . The carrier frequency is

. At this frequency, path loss in dB is given byin the case where , where

is the distance in kilometers between the BS and the user. Inthe case , . The signalbandwidth is equal to 5 MHz and the thermal noise powerspectral density is equal to . Each cell hasa radius .

Asymptotically Optimal Pivot-Distance and Frequency ReuseFactor: We first apply the results of Sections IV and VI in orderto obtain the values of the asymptotically optimal pivot-dis-tances and the asymptotically optimal reuse factor .These values are necessary for the implementation of the sim-plified allocation algorithm proposed in Section III. Each ofthe two cells is assumed to have in the asymptotic regime thesame uniform distribution of users: where

. The average rate requirement in each cell isassumed to be the same, too: , where is de-fined in Section IV-A-2 as the average data rate in Cell mea-sured in bits/sec/Hz. In this case, the optimal pivot-distance isthe same in each cell i.e., . Define .The value of and was obtained using the method de-picted by Section IV-C and Section VI respectively. Denote by

the total data rate of all the users of a sector measured inbits/sec . Fig. 2 and Fig. 3 plot respectivelyand the normalized pivot-distance as functions of thetotal rate for two values of the path loss exponent: and

. Note from Fig. 2 that and are both decreasingfunctions of . This result is expected, given that higher valuesof will lead to higher transmit powers and consequently tohigher levels of interference. More users will need thus to be“protected” from the higher interference. For that purpose, thepivot-position must be closer to the base station and a larger partof the available bandwidth must be reserved for the protectedbands and . Note also that, in the case , “less pro-tection” is needed than in the case where . In other words,

and .This observation can be explained by the fact that, when the pathloss exponent is higher, the interference produced by the adja-cent base station will undergo more fading than in the case whenthe path loss exponent is lower.

Simplified Resource Allocation: In Section III, we proposeda suboptimal allocation algorithm characterized by its reducedcomputational complexity compared to the optimal allocationalgorithm depicted in [1]. This algorithm assumes fixed pivot-distances , . Here, we study the performance of



Fig. 2. Optimal reuse factor versus sum rate.

Fig. 3. Optimal pivot-distance versus sum rate.

this algorithm when and are chosen accordingto the procedure provided in Section IV-C, i.e.,and , where is the asymptotically optimalpivot-distance defined earlier in this section. In order to studythe performance of this algorithm, we need to compare, fora large number of system settings, the total transmitpower that must be spent when applying the simplified algo-rithm, with the total transmit power that must be spentwhen the optimal resource allocation scheme of Part I [1] isapplied. The results must then be averaged in order to obtainperformance measurements that are independent of the partic-ular system setting. We consider therefore that users in eachcell are randomly distributed and that the distance separatingeach user from the base station is a random variable with a uni-form distribution on the interval . On the other hand, weassume without restriction that all users have the same targetrate, and that the number of users is the same for the two cells

. Define as the vector containing the positions of allthe users in the system i.e., . Re-call that is a random variable with a uniform distributionon . For each realization of , define as thetotal transmit power that results from applying the optimal jointresource allocation scheme of Part I with the value of the reusefactor fixed to . Define . In thesame way, denote by the total transmit power thatresults from applying the simplified resource allocation schemeof Section VI with the value of the reuse factor fixed todefined in Section VI. For each realization of the random vector

, the values of and were calculated and

Fig. 4. Optimal and suboptimal transmit power versus sum rate.

Fig. 5. Transmit power versus the pivot-distance � for the simplified allocationscheme (� � �� , � � ��).

then averaged to obtain and re-

spectively. We plot in Fig. 4 the values of andfor a range of values of the sum rate mea-

sured in bits/sec in two cases:and . The error bars in the figure represents the vari-ance of the random variable in the case .In the same figure, the corresponding values of the asymptotictransmit power defined by Theorem 1 are also plotted. Thisfigure shows that, even for a reasonable number of users equalto 25 in each cell, the transmit power needed when we apply thesuboptimal algorithm is very close to the power needed when weapply the optimal resource allocation scheme. The gap betweenthe two powers is of course even smaller for . This re-sult validates Proposition 2 which states that our proposed sub-optimal resource allocation scheme is asymptotically optimal.Fig. 5 is dedicated to illustrate the sensitivity of the simplified al-location scheme with respect to the pivot-distance in thecase . For that sake, the figure plots the total transmitpower resulting from applying the simplified scheme as a func-tion of . The minimum in the figure corresponds to theasymptotically optimal pivot distance . We notethat using values different from increases the suboptimalityof the simplified scheme. Let us go back to Fig. 4. The latterfigure shows that over the range of the considered values ofthe total data rate , the total transmit power for

is practically equal to the asymptotic power . This



Fig. 6. � �� versus number of users per cell.

result suggests that, for a number of users equal to 50 in eachcell, the system is already in its asymptotic regime. In orderto validate the latter affirmation, one still needs to investigate

the value of the mean square error as well. This

is done by Fig. 6, which plots , themean square error normalized by .

VIII. CONCLUSION

In this pair of papers, the resource allocation problem forsectorized downlink OFDMA systems has been studied in thecontext of a partial reuse factor . In the first part ofthis work, the general solution to the (nonconvex) optimizationproblem has been provided. It has been proved that the solu-tion admits a simple form and that the initial tedious problemreduces to the identification of a limited number of parameters.As a noticeable property, it has been proved that the optimalresource allocation policy is “binary”: there exists a pivot-dis-tance to the BS such that users who are farther than this distanceshould only modulate protected subcarriers, while closest usersshould only modulate reused subcarriers. A resource allocationalgorithm has been also proposed.

In the second part, we proposed a suboptimal resource al-location algorithm which avoids the costly search for param-eters such as the optimal pivot-distance. In the proposed pro-cedure, the optimal pivot-distance is simply replaced by a fixedvalue. In order to provide a method to relevantly select this fixedpivot-distance, the asymptotic behavior of the optimal resourceallocation has been studied as the number of users tends to in-finity. In the case where the fixed pivot-distance associated withthe simplified algorithm is chosen to be equal to the asymptoti-cally optimal pivot-distance, it has been shown that our simpli-fied resource allocation algorithm is asymptotically equivalentto the optimal one as the number of users increases. Simulationsproved the relevancy of our algorithm even for a small number

of users. Using the results of the asymptotic study, the optimalvalue of the reuse factor has been characterized. It is defined asthe value of which minimizes the asymptotic value of the min-imum transmit power. Our simulations proved that substantialimprovements in terms of spectral efficiency can be expectedwhen using the relevant value of the reuse factor.

APPENDIX APROOF OF THEOREM 1

Theorem 1 characterizes the asymptotic behavior of theminimal transmit power resulting from applying the optimalresource allocation when the number of users tends toinfinity. It is thus useful at this point to recall the theorem givenin the first part of this work which characterizes the optimalallocation for finite values of . For each celland for each , define by and the uniquepositive numbers such that and

, with byconvention.

Theorem 2 [1]:(A) Any global solution to the joint resource allocation

problem has the following form. For each Cell , thereexists an integer , and there exist fourpositive numbers such that1) for each ,

(23)

2) for each ,

(24)

3) for [see the equation at the bottom of thepage].

(B) For each , the system formed bythe following four equations is satisfied:

(26)

(27)

(28)

(29)

(25)



where the values of and in (29) are the functionsof defined by (23).

(C) Furthermore, for each and for any arbitraryvalues and , the system of equationsadmits at most one solution .

In Section IV-A-2, we obtained that for each cell ,

(30)

(31)

where andand where is the pivot-distance

i.e., the position of user . Our aim is to prove thatconverges as , and to

characterize the limit. For each cell , sequenceis bounded by definition . Consider

a subsequence such that converges toa certain limit, say . We prove that in this case, allquantities convergeto some values which we shall characterize.Focus for instance on sequence . Recalling thattends to zero as and replacing

each with expression (23) , weobtain immediately

(32)

where we defined

(33)

for each . In the asymptotic regime, we obtain the fol-lowing lemma.

Lemma 2: As , sequence converges to theunique solution to the following equation:

(34)

Proof: Existence and uniqueness of the solution to (34) isstraightforward since function is strictly de-creasing from to 0 on . We remark that sequenceis bounded i.e., for a certain constant . In order toprove this claim, assume that there exists a subsequencewhich converges to infinity. This hypothesis implies that thesubsequence given by the LHS of (32) for of the form

converges to zero as . This is in contradic-tion with (32) which states that the latter sequence converges to

. Using similar arguments, it can be shown that

is lower bounded by a certain , i.e., . De-note by any accumulation point of and definea subsequence of (i.e., coincides with for acertain function ) which converges to . We prove that isgiven by (34). Define . We show thatthe difference

tends to zero as . By the triangular inequality

Respectively, denote by and the first,second, and third terms of the above equation. We firststudy . Clearly, function is bounded on

. Denote by an upper bound. Then,, where

(or if ). Re-call that converges to by definition, so that

converges to zeroas long as measure has no mass point at . Sinceconverges weakly to , it is straightforward to show that

, and thus , tend to zero. Now focus on. The first term which

composes converges to bythe weak convergence of to . The second term

converges to the same limit byLebesgue’s dominated convergence Theorem. Thus,tends to zero. In order to prove that tends to zero,we remark that , where thesupremum is taken w.r.t. .Denote by the latter supremum. We easily obtain

, sothat .Since is a probability measure,

. Thus, tends to zero as tends to infinity.Putting all pieces together, tends to zero. Using (32),



converges to

. By continuity arguments,satisfies (34). Thus, is a bounded sequence such thatany accumulation point is equal to defined by (34). Thus,

.

Using Lemma 2, we may now characterize the limit of(31) as . Using the fact that and

along with some technical arguments whichare similar to the ones used in the proof of Lemma 2, we obtain

(35)where is the unique solution to (34). As convergesweakly to , converges to

(36)

The same approach can be used to analyze the be-havior of sequences and for each

. After similar derivations, we ob-tain the following result. As , sequence

converges to the unique solution tothe following system of six equations:

(37)where and are the limits of and respec-tively. We discuss now the existence and the uniqueness of thesolution to the above system of equation. For that sake, recallthe definition of functions and given by (12) and (33) re-spectively. Note that ,and that . Define

for . By applying this new notation, Thefirst two equations of system (37) give place to the followingsystem of four equations:

(38)The existence and the uniqueness of the solutionto the system (38) was thoroughly studied in [2]. Ap-plying the results of [2] in our context, we conclude that

is unique. Weturn now back to the third equation of system (37) to get thefollowing equality:

The latter equation proves the uniqueness of for .The uniqueness of follows directly from the same equation.

So far, we have proved the uniqueness of the solution to thesystem (37) of equation. As for the convergence of sequences

tothis unique solution, its proof is omitted here due to the lack ofspace, but follows the same ideas as the proof of convergence of

and provided above.We have thus managed to prove that for any convergent

subsequence , the set of pa-rametersconverges to some values which are com-pletely characterized by the system of (34), (36) and (37), asfunctions . Using decomposition , thesystem formed by (34), (36) and (37) is equivalent to the system(19), (20), (21), (22) provided in Theorem 1. At this point, wethus proved that at least for some subsequences defined asabove, the subsequence converges to a limit which hasthe form given by Theorem 1. The remaining task is to provethat is a convergent sequence.

First, note that is a bounded sequence. Indeed, isdefined as the minimum power that can be transmitted by thenetwork to satisfy the rate requirements. By definition, isthus less than the power obtained when using the naive solutionwhich consists in forcing each base station to transmit only inthe protected band ( is forced to zero for each user of eachcell ). Now it can easily be shown that when , thepower associated with this naive solution converges to a con-stant. As a consequence, one can determine an upper-bound on

which does not depend on .Second, assume for instance that and are two accu-

mulation points of sequence . By contradiction, assumethat . Extract for instance a certain subsequence of

which converges to . Inside this subsequence, one canfurther extract a subsequence, say , such that

where and are some constants both (just use the fact thatis bounded for each ). Clearly, can be written as in

(18), where parameters satisfy the system of(19), (20), (21), (22). We now consider the following suboptimalresource allocation policy for finite numbers of users and

. In each cell , users whose distance totheir BS is less than are forced to modulate in the interferenceband only, while users which are farther than are forcedto modulate in the protected band only. In other words, foreach user in cell , we impose

(39)

Particular values of the (nonzero) resource allocation param-eters are obtained by minimizing the classical jointmulticell resource allocation problem (Problem 2 in [1]), onlyincluding the additional constraint . As a new constraint hasbeen added, it is clear that the total power transmitted by thenetwork, say , is always larger than the total power



achieved by the optimal resource allocation, for any . On theother hand, using the same asymptotic tools as previously, it canbe shown after some algebra that

In other words, this suboptimal solution performs as good asthe optimal one when has the form for some .Although we omit the proof, this observation is rather intuitive.Indeed for such , the optimal values of the pivot-dis-tances converge to the arbitrary ones . Even more im-portantly, it can be shown that the total power spentwhen using the suboptimal procedure converges as .Therefore,

Now consider a subsequence such that, and compare our suboptimal allocation policy

to the optimal one for the ’s of the form . As, there exist a certain and

there exists a certain such that for any ,

The above inequality contradicts the fact that is theglobal solution to the joint multicell resource allocationproblem (Problem 2 in [1]). Therefore, necessarily coin-cides with . This proves that converges to . Tocomplete the proof of Theorem 1, one still needs to prove thatfor any fixed value of , the system formed by(19), (20), (21), (22) admits at most one solution. The mainideas of this proof were evoked in the proof of Proposition 1of [1]. However, the complete proof is omitted due to lack ofspace.

REFERENCES

[1] N. Ksairi, P. Bianchi, P. Ciblat, and W. Hachem, “Resource allocationfor downlink sectorized cellular OFDMA systems: Part I—Optimal al-location,” IEEE Trans. Signal Process., vol. 58, no. 2, pp. 720–734,Feb. 2010.

[2] S. Gault, W. Hachem, and P. Ciblat, “Performance analysis of anOFDMA transmission system in a multi-cell environment,” IEEETrans. Commun., vol. 55, no. 12, pp. 2143–2159, Dec. 2005.

[3] J. Chen, R. A. Berry, and M. L. Honig, “Asymptotic analysis of down-link OFDMA capacity,” presented at the 44th Annu. Allerton Conf.,Monticello, IL, Sep. 2006.

[4] H. Wang and B. Chen, “Asymptotic distributions and peak power anal-ysis for uplink OFDMA signals,” in Proc. IEEE Int. Conf. Acoust.,Speech, Signal Process., May 2004, vol. 4, pp. 1085–1088.

[5] E. Biglieri, G. Caire, G. Tarrico, and E. Viterbo, “How fading affectsCDMA: An asymptotic analysis with linear receivers,” IEEE J. Sel.Areas Commun., vol. 19, no. 2, pp. 191–201, Feb. 2001.

[6] D. Gesbert and M. Kountouris, “Joint power control and user sched-uling in multi-cell wireless networks: Capacity scaling laws,” IEEETrans. Inf. Theory, Sep. 2007 [Online]. Available: http://arxiv.org/abs/0709.2851, submitted for publication

[7] P. Billingsley, Probability and Measure, 3rd ed. New York: Wiley,1995.

[8] M. Maqbool, M. Coupechoux, and P. Godlewski, “Comparison ofvarious frequency reuse patterns for WiMAX networks with adaptivebeamforming,” in IEEE Veh. Technol. Conf. (VTC), Singapore, May2008, pp. 2582–2586.

[9] H. Jia, Z. Zhang, G. Yu, P. Cheng, and S. Li, “On the performanceof IEEE 802.16 OFDMA system under different frequency reuse andsubcarrier permutation patterns,” in Proc. IEEE Int. Commun. Conf.(ICC), Jun. 2007, pp. 5720–5725.

[10] F. Wang, A. Ghosh, C. Sankaran, and S. Benes, “WiMAX system per-formance with multiple transmit and multiple receive antennas,” inProc. IEEE Veh. Technol. Conf. (VTC), Apr. 2007, pp. 2807–2811.

[11] T. Thanabalasingham, S. V. Hanly, L. H. Andrew, and J. Papandri-opoulos, “Joint allocation of subcarriers and transmit powers in a mul-tiuser OFDM cellular network,” in Proc. IEEE Int. Conf. Commun.(ICC), Jun. 2006, vol. 1, pp. 269–274.

[12] R. D. Yates, “A framework for uplink power control in cellular radiosystems,” IEEE J. Sel. Areas Commun., vol. 13, no. 7, pp. 1341–1347,Sep. 1995.

[13] N. Ksairi, P. Bianchi, P. Ciblat, and W. Hachem, “Resource alloca-tion for downlink cellular OFDMA Systems,” Tech. Rep., May 2009[Online]. Available: http://www.tsi.enst.fr/~bianchi/Resource_alloca-tion_technical_report.pdf

[14] J. F. Bonnans and A. Shapiro, Perturbation Analysis of OptimizationProblems. New York: Springer, 2000.

[15] COST Action 231, “Digital mobile radio toward future generation sys-tems,” European Communities, Final Rep., Tech. Rep. EUR 18957,1999.

Nassar Ksairi (S’08–M’09) was born in Damascus,Syria, in 1982. He received the M.Sc. degree inwireless communication systems from the ÉcoleSupérieure d’Électricité (Supélec), Gif-sur-Yvette,France, in 2006. He is currently working towards thePh.D. degree at the Telecommunication Departmentof Supélec.

His research interests include resource allocationfor wireless networks (distributed allocation algo-rithms, allocation with base station cooperation andmacro diversity), and cooperative networks (relaying

protocols and their performance analysis).

Pascal Bianchi (M’06) was born in Nancy, France,in 1977. He received the M.Sc. degree from Supélec-Paris XI in 2000 and the Ph.D. degree from the Uni-versity of Marne-la-Valé in 2003.

From 2003 to 2009, he was an Associate Professorat the Telecommunication Department of Supélec. In2009, he joined the Signal and Image Processing De-partment of Telecom ParisTech, Paris, France. His re-search interests are in the area of statistical signal pro-cessing for sensor networks and multiuser commu-nication systems. They include distributed detection

and estimation, resource allocation strategies, applications of random matrixtheory, relay networks, and blind signal processing.

Philippe Ciblat (M’06) was born in Paris, France,in 1973. He received the Engineering degree fromthe Ecole Nationale Supérieure des Télécommunica-tions (ENST) and the D.E.A. degree (french equiv-alent to the M.Sc. degree) in automatic control andsignal processing from the Université de Paris-Sud,Orsay, France, both in 1996, and the Ph.D. degreefrom Laboratoire Système de Communication of theUniversité de Marne-la-Vallée, France, in 2000. HisPh.D. dissertation focused on the estimation issue innoncooperative telecommunications.

In 2001, he was a Postdoctoral Researcher with the Laboratoire des Télécom-munications, Université Catholique de Louvain, Belgium. At the end of 2001,he joined the Département de Communications et Electronique at ENST, Paris,France, as an Associate Professor. His research areas include statistical and dig-ital signal processing (blind equalization, frequency estimation, and asymptoticperformance analysis) and signal processing for digital communications (syn-chronization for OFDM modulations and the CDMA scheme, access techniqueand localization for UWB, and the design of global systems).

Dr. Ciblat served as Associate Editor for the IEEE COMMUNICATIONS

LETTERS from 2004 to 2007. Since 2008, he has served as Associate Editor forthe IEEE TRANSACTIONS ON SIGNAL PROCESSING.



Walid Hachem (M’04) was born in Bhamdoun,Lebanon, in 1967. He received the Engineeringdegree in telecommunications from St. JosephUniversity (ESIB), Beirut, Lebanon, in 1989,the Master’s degree from Telecom ParisTech(then Ecole Nationale Supérieure des Télécom-munications, ENST), France, in 1990, the Ph.D.degree in signal processing from the Universitéde Marne-La-Vallée in 2000 and the Habilitationà Diriger des Recherches from the Université deParis-Sud, Orsay, France, in 2006.

Between 1990 and 2000, he was a Signal Processing Engineer with Philipsand then with FERMA. In 2001, he joined the Ecole Supérieure d’Electricité(Supélec). Since 2007, he has been a CNRS (Centre national de la RechercheScientifique) researcher at Telecom ParisTech. His research interests concern theanalysis of communication systems based on the study of large random matrices,detection and estimation, and cooperative wireless networks.

Dr. Hachem currently serves as an Associate Editor for the IEEETRANSACTIONS ON SIGNAL PROCESSING.


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